mnb96
Jun4-11, 11:31 AM
Hello,
today I was thinking the following: it is known that Clifford Algebras provide a general framework to "construct" and generalize the algebras of real numbers, complex numbers and quaternions. Clifford Algebras can also be used to nicely generalize some geometrical concepts to n-dimensions. However, unless I am wrong, Clifford Algebras fail to encapsulate octonions (due to their non-associativity), which represent the last of the four possible normed division algebras of the reals. This gave rise to two questions:
1) Clifford Algebras primarily rely upon the axiomatic properties of the geometric product, which is defined to be associative. Is it possible that there exist a "more general geometric-product", by means of which we might be able to encapsulate both Clifford Algebras and octonions?
2) complex numbers and quaternions can be used to describe many geometrical concepts (e.g. rotations). Can octonions describe other more involved geometrical concepts that cannot be modeled with quaternions? If so, which ones?
Thanks in advance.
today I was thinking the following: it is known that Clifford Algebras provide a general framework to "construct" and generalize the algebras of real numbers, complex numbers and quaternions. Clifford Algebras can also be used to nicely generalize some geometrical concepts to n-dimensions. However, unless I am wrong, Clifford Algebras fail to encapsulate octonions (due to their non-associativity), which represent the last of the four possible normed division algebras of the reals. This gave rise to two questions:
1) Clifford Algebras primarily rely upon the axiomatic properties of the geometric product, which is defined to be associative. Is it possible that there exist a "more general geometric-product", by means of which we might be able to encapsulate both Clifford Algebras and octonions?
2) complex numbers and quaternions can be used to describe many geometrical concepts (e.g. rotations). Can octonions describe other more involved geometrical concepts that cannot be modeled with quaternions? If so, which ones?
Thanks in advance.